In this paper, we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding H 1 0 (Ω) ֒→ L p (Ω) on bounded convex domain in R 2. We estimate the best constant by computing the corresponding extremal function using a verified numerical computation. Verified numerical inclusions of the best constant on a square domain are presented.

In this paper, we propose a method for estimating the Sobolev-type embedding constant from W 1,q ( ) to L p ( ) on a domain ⊂ R n (n = 2, 3, . . . ) with minimally smooth boundary (also known as a Lipschitz domain), where p ∈ (n/(n -1), ∞) and q = np/(n + p). We estimate the embedding constant by constructing an extension operator from W 1,q ( ) to W 1,q (R n ) and computing its operator norm. We also present some examples of estimating the embedding constant for certain domains. MSC: 46E35

Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite-dimensional Newton-type fixed point equation $$w = - {\mathcal {L}}^{-1} {\mathcal {F}}(\hat{u}) + {\mathcal {L}}^{-1} {\mathcal {G}}(w)$$ w = - L - 1 F ( u ^ ) + L - 1 G ( w ) , where $${\mathcal {L}}$$ L is a linearized operator, $${\mathcal {F}}(\hat{u})$$ F ( u ^ ) is a residual, and $${\mathcal {G}}(w)$$ G ( w ) is a nonlinear term. Therefore, the estimations of $$\Vert {\mathcal {L}}^{-1} {\mathcal {F}}(\hat{u}) \Vert $$ ‖ L - 1 F ( u ^ ) ‖ and $$\Vert {\mathcal {L}}^{-1}{\mathcal {G}}(w) \Vert $$ ‖ L - 1 G ( w ) ‖ play major roles in the verification procedures . In this paper, using a similar concept to block Gaussian elimination and its corresponding ‘Schur complement’ for matrix problems, we represent the inverse operator $${\mathcal {L}}^{-1}$$ L - 1 as an infinite-dimensional operator matrix that can be decomposed into two parts: finite-dimensional and infinite-dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, which enables a more efficient verification procedure compared with existing Nakao’s methods for the solution of elliptic PDEs. We present some numerical examples that confirm the usefulness of the proposed method. Related results obtained from the representation of the operator matrix as $${\mathcal {L}}^{-1}$$ L - 1 are presented in the “Appendix”.

This paper is concerned with an explicit value of the embedding constant from to for a domain (), where . We previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein’s extension operator. Although this formula can be applied to a domain Ω that can be divided into a finite number of Lipschitz domains, there was room for improvement in terms of accuracy. In this paper, we report that the accuracy of the embedding constant is significantly improved by restricting Ω to a domain dividable into bounded convex domains.

In this paper, we propose a numerical method for verifying the positiveness of solutions to semilinear elliptic boundary value problems. We provide a sufficient condition for a solution to an elliptic problem to be positive in the domain of the problem, which can be checked numerically without requiring a complicated computation. Although we focus on the homogeneous Dirichlet case in this paper (in fact, it is often possible that solutions are not positive near the boundary in this case), our method can be applied naturally to other boundary conditions. We present some numerical examples.

Improved componentwise error bounds for approximate solutions of linear systems are derived in the case where the coefficient of a given linear system is an H -matrix. One of the error bounds presented in this paper proves to be tighter than the existing error bound, which is effective especially for ill-conditioned cases. Numerical experiments are performed to illustrate the effect of the improvements.

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